Risk Management puzzles, solved step by step

Puzzles: 100
Traced to a firm: 17
Topics: 13
Hard: 30

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050An EWMA volatility model with a decay factor of 0.94 had yesterday's daily volatility estimate at 1.2%. Today's return is minus 3%. What is the updated volatility estimate?Statistics and estimationCoreBank market riskQuant risk
Try it first
Roughly where does the new daily volatility land?
Show the worked solution
About 1.38%. EWMA updates the variance, not the volatility: new variance is 0.94 x 1.2 squared plus 0.06 x 3 squared, which is 1.3536 plus 0.54, or 1.8936. The square root is 1.376%. The shock has only a 6% weight but enters squared, so it supplies 29% of the new variance.
Why blend variances rather than volatilities?
A household tracking how much its grocery bill swings would be misled if it averaged the size of the swings in rupees but ignored that one big swing matters far more than several small ones. Variance is the average squared move, so the model updates squared returns, and a move 2.5 times normal size counts 6.25 times as much. An EWMAExponentially weighted moving average: each day the estimate keeps a fixed share of yesterday and adds the rest from today, so older days fade geometrically. of variances is what makes one large shock move the estimate quickly. Blending the volatilities directly gives 1.31%, understating the jump.
Today's minus 3% return carries only a 6% weight in the EWMA update, but because it enters squared it supplies 28.5% of the new variance of 1.8936, lifting daily volatility from 1.20% to 1.38%.
The relationship
$\sigma_t^2 = \lambda\,\sigma_{t-1}^2 + (1-\lambda)\,r_t^2 = 0.94(1.44) + 0.06(9) = 1.8936, \quad \sigma_t = 1.376\%$
\lambda the decay factor, 0.94
\sigma_{t-1} yesterday's volatility estimate, 1.2%
r_t today's return, minus 3%
What it says in wordsKeep 94% of yesterday's variance, add 6% of today's squared return, then take the square root.
What happens next, and what are the model's limits?
If tomorrow is flat, the estimate decays to the square root of 0.94 x 1.8936, about 1.33%. Each day of calm keeps 94% of the variance, so a shock's influence halves in about 11 trading days: ln 0.5 over ln 0.94. That is the design choice behind 0.94: fast enough to react to a new regime within days, slow enough not to swing on every move.
The limits: EWMA has no pull towards a long-run average, so after a calm spell it can drift very low and understate risk just before volatility returns, which GARCH-type models address with a mean-reversion term. It also treats up and down moves alike, while falling markets often raise volatility more than rising ones. And the choice of 0.94 is a convention for daily data, not a law, so the estimate should be backtested against realised moves.
Where candidates lose it
The usual slip is to blend the two volatilities directly, 0.94 x 1.2 plus 0.06 x 3, and answer 1.31%. It looks like the formula but applies it to the wrong quantity and understates the effect of the shock.
The second is forgetting the square root at the end and quoting 1.89% as a volatility. Say the units at each step: variance in, variance out, then volatility.
What the interviewer asks next
What would the estimate be with lambda of 0.97 instead?
How many quiet days until the estimate is back below 1.25%?
Why might a risk manager prefer GARCH to EWMA for a ten-day VaR?
052An asset has an arithmetic average return of 10% a year and a volatility of 30%. Roughly what is its compound annual growth rate?Compounding and drawdownsCoreAsset manager risk
Try it first
Pick the closest before you calculate.
Show the worked solution
About 5.5% a year. Compound growth is roughly the arithmetic mean minus half the variance. The variance is 0.30 squared, 0.09, and half of it is 4.5 points, so 10% becomes about 5.5%. A quick check: +40% then -20% averages 10% with a 30% spread, and Rs 100 ends at Rs 112, which is 5.8% a year.
Why is the average return not the growth rate?
A shopkeeper who marks a shirt up 40% and later cuts it 20% has not made 20% on the pair of moves. He has made 12%, because the cut is taken from the higher price. Returns work the same way. A loss is always taken from whatever the gain left you, so ups and downs of the same average size leave you with less than steady growth at that average. The more the returns swing, the bigger the shortfall.
Rs 100 rising 40% and then falling 20% ends at Rs 112, a compound 5.8% a year, even though the two returns average 10%. The rule of thumb puts compound growth at the 10% average less half the variance of 0.09, which is about 5.5%.
Where does half the variance come from, and how good is it?
Take logs. The log of one plus a return is roughly the return minus half its square, so averaging the logs knocks off about half the variance. The drag grows with the square of volatility: at 15% volatility it is about 1.1 points, at 30% it is 4.5. The rule is an approximation that works best for small returns. The two-year example lands at 5.8%, and other return patterns with the same average and spread give slightly different answers.
The relationship
$g \approx \mu - \frac{\sigma^2}{2} = 0.10 - \frac{0.30^2}{2} = 0.055$
g the compound annual growth rate
\mu the arithmetic average annual return, 10%
\sigma the annual volatility, 30%
What it says in wordsCompound growth is the average return less half the variance.
For a risk manager this is why two funds with the same average return are not the same fund. The one with double the volatility has four times the drag, and its investors end up with less money even though the average looks identical.
Where candidates lose it
The trap is answering 10% because the question gave you 10%. The interviewer is checking whether you know that averages of returns overstate what an investor compounds, and that the gap is driven by volatility.
The second slip is subtracting the whole variance, or subtracting half the volatility, and landing at 1% or -5%. Square first, then halve: 0.09 over 2 is 4.5 points.
What the interviewer asks next
What volatility would make the compound growth zero with a 10% average?
If you lever this asset 2x, what happens to the average and to the compound growth?
Which number should a fund report to investors, and why?
053A hedge instrument has a correlation of 0.8 with your position. If you put on the best possible hedge, what share of the position's variance does it remove, and how much of the volatility is left?Correlation and diversificationCoreBank market riskQuant risk
Try it first
Your gut first: how much of the volatility does a 0.8 correlated hedge leave behind?
Show the worked solution
The best hedge removes 64% of the variance and leaves 60% of the volatility. With the minimum variance hedge ratio, the share of variance removed is the correlation squared, 0.8 squared or 0.64. That leaves 36% of the variance, and because volatility is its square root, 60% of the original volatility is still there.
Why does a correlation of 0.8 leave so much behind?
Two friends walk home along roughly the same road. Most of the time they are close, but each takes a detour now and then, and the gap between them on those days is what a hedge cannot touch. A hedge only cancels the part of your position that moves with the instrument. The part it cancels is rho squared of the variance, and whatever is left, called basis riskThe risk that the hedge and the position do not move together, so the hedge gains or loses a different amount from the position., is all yours. At 0.8 that residual is 36% of variance.
A hedge with a 0.8 correlation removes 64 of every 100 units of variance and leaves 36. Because volatility is the square root of variance, the same hedge leaves 60 of every 100 units of volatility, so most of the swing you feel day to day survives.
How do you get from variance left to volatility left?
Take the square root. Risk managers quote variance when they add risks and volatility when they talk about losses, and the square root between them is where candidates lose the answer. A hedge that removes 64% of variance sounds impressive; saying it leaves 60% of the volatility is the honest version, and it is the number that matters for a VaR limit, which scales with volatility.
The relationship
$\sigma_{\text{hedged}} = \sigma_P\sqrt{1-\rho^2} = \sigma_P\sqrt{1-0.64} = 0.6\,\sigma_P$
\sigma_P the volatility of the unhedged position
\rho the correlation between position and hedge, 0.8
What it says in wordsThe best hedge leaves the position's volatility times the square root of one minus the correlation squared.
The limitation: the 0.8 was measured on past data. In a stress the correlation can fall, and the residual grows just when you need the hedge most. That is why desks watch the stability of a hedge correlation as closely as its level.
Where candidates lose it
The fast wrong answer is 20% of the risk left, one minus the correlation. It treats correlation as the share of risk removed, which it is not. Squaring the correlation gives the share of variance explained.
The second trap is stopping at 36% and forgetting that volatility is the square root. Give both numbers, 64% of variance removed and 60% of volatility left, and say which one a VaR limit sees.
What the interviewer asks next
What correlation do you need to cut volatility in half?
What hedge ratio achieves this if the position has twice the hedge instrument's volatility?
Why might a proxy hedge's correlation drop in a crisis?
059Failed trades on a settlement desk average 12 a day with a standard deviation of 3. An amber alert fires at 18. If nothing about the process has changed, how many false amber alerts should you expect in a 250-day year?Operational loss and fraudCoreOperational risk
Try it first
Your instinct first: how many false ambers a year?
Show the worked solution
About 6 false ambers a year. The threshold of 18 is two standard deviations above the mean of 12. If daily fails are roughly normal, about 2.3% of days land above two standard deviations on the high side, and 2.28% of 250 days is 5.7. Roughly one quiet day in every 44 will trip the alert with nothing wrong.
Why does a sensible threshold still fire when nothing is wrong?
A smoke alarm set sensitive enough to catch every real fire also goes off when someone burns toast. Set it deaf enough never to react to toast, and it may miss a real fire. Operational alerts face the same trade. Any threshold drawn on a noisy count carries a false alarm rate you can compute in advance, and the only way to cut it is to accept missing more real problems.
Daily fails centred on 12 with a standard deviation of 3 put the amber line at 18, two standard deviations up. The shaded tail holds 2.3% of days, which over a 250-day year means about 5.7 false ambers with the process unchanged.
How precise is the answer of about six?
It rests on two assumptions worth saying. First, fails are whole numbers: if a count of exactly 18 triggers the alert, a continuity correction moves the tail to about 3.3% and the count to about 8. Second, real operational counts often have fatter tails than a bell curve, with bad days clustering around month ends and system changes. Both push the true false alarm count up, so six is a floor, not a ceiling.
The relationship
$z = \frac{18-12}{3} = 2 \qquad P(Z>2) \approx 2.3\% \qquad 0.023 \times 250 \approx 5.7$
z how many standard deviations the threshold sits above the mean
P(Z>2) the share of a normal distribution beyond two standard deviations on one side
What it says in wordsConvert the threshold to standard deviations, read off the tail, and multiply by the number of days.
In the room, close with what you would do: tell the operations team to expect about one amber every two months from noise alone, so that a cluster of ambers in one week, not a single one, is what triggers an investigation.
Where candidates lose it
The trap is answering none, or close to none, because 18 looks far from 12. Six fails is only two standard deviations, and two standard deviations is not rare over 250 tries.
The second slip is using the two-sided 5% and getting 12 or 13 a year. Only the high side triggers an amber, so the tail is about 2.3%, not about 5%.
What the interviewer asks next
Where would you set the threshold to get about one false amber a year?
Three ambers arrive in one week. How surprised should you be if nothing has changed?
Why might failed trade counts not follow a normal distribution?
060A one-year European call and put on a non-dividend stock, both struck at 100, trade at 12 and 7. The stock is at 100. What interest rate does put-call parity imply?Options and GreeksCoreBank market risk
Try it first
Which relationship do you use?
Show the worked solution
About 5.26% a year with annual compounding, or 5.13% continuously compounded. Parity says call minus put equals stock minus the present value of the strike. Here 12 minus 7 is 5, so the present value of 100 must be 95. A one-year discount factor of 0.95 means 100 over 95 minus 1, about 5.26%.
Why must call minus put equal the stock minus the discounted strike?
Agreeing today to buy a house next year at a fixed price is the same as buying it now with money borrowed until then: either way you own the house next year and pay the fixed price. A long call plus a short put at the same strike is that agreement. At expiry the pair pays the stock minus the strike in every state, so today it must cost the same as owning the stock and owing the strike in a year, which is S minus the present value of K.
A long call and a short put struck at 100 together pay the stock price minus 100 in every outcome. Call 12 minus put 7 is 5, which must equal the stock at 100 minus the present value of the strike, so that present value is 95 and the implied rate is 5.26% a year.
Why would a risk manager care about the rate hidden in option prices?
Because it is a check that comes free. If the rate implied by parity sits far from the funding rate the desk actually pays, either the marks are stale, a dividend has been missed, or the options are American and parity no longer holds exactly. Model validation teams run exactly this test on option books to catch mispriced marks before they show up as a loss.
The relationship
$C - P = S - \frac{K}{1+r} \;\Rightarrow\; 5 = 100 - \frac{100}{1+r} \;\Rightarrow\; r = \frac{100}{95} - 1$
C, P the call and put prices, 12 and 7
S the stock price, 100
K the common strike, 100
r the one-year interest rate implied by the prices
What it says in wordsThe gap between an at-the-money call and put is the interest on the strike, so the prices reveal the rate.
The limitation: parity holds exactly only for European options on a stock paying no dividend before expiry. A dividend would lower the stock's forward and push the implied rate the other way, so state the assumption before you give the number.
Where candidates lose it
The common error is to write 5 over 100 and answer 5%. That treats 5 as the interest on 100, but 5 is what you save today, and the rate is measured on the 95 you actually pay. 100 over 95 minus 1 is 5.26%.
The other trap is forgetting the conditions. Say European, no dividends, and one year, then give the number; an interviewer will often follow up with a dividend to see if you adjust.
What the interviewer asks next
The stock pays a dividend of 2 in six months. What rate is implied now?
The call trades at 13 with the put unchanged. What trade locks in a profit?
Why does parity not hold exactly for American options?
061A loan book is 60% retail loans with a 3% default rate and 40% corporate loans with a 1% default rate. A loan picked at random has defaulted. What is the probability it was a retail loan?Probability and base ratesCoreBank credit riskRisk GCC
Try it first
Which number is closest?
Show the worked solution
About 81.8%. Retail defaults make up 60% x 3% = 1.8% of the book and corporate defaults 40% x 1% = 0.4%, so 2.2% of all loans default. Given that a loan is among the 2.2%, the chance it is retail is 1.8 divided by 2.2, which is 9 in 11, or about 82%.
Why is the answer not simply 60%?
Suppose 60% of the cars on a road are small hatchbacks and 40% are trucks, but trucks are far more likely to have a broken tail light. If a policeman stops a car for a broken tail light, it is more likely to be a truck than the 40% share suggests. The stop tells you something. Learning that a loan defaulted is evidence, and it shifts the odds towards the segment that defaults more often.
Retail loans are 60% of the book and default 3% of the time, contributing 1.8% of all loans as defaults; corporate loans contribute 0.4%. Of the 2.2% that default, 1.8 is retail, so a defaulted loan is retail with probability 81.8%.
What is the quickest way to do this in your head?
Use 1,000 loans. There are 600 retail loans, of which 18 default, and 400 corporate loans, of which 4 default. Out of 22 defaults, 18 are retail, so the answer is 18 over 22, and natural numbers make the base rate impossible to forget. This is Bayes' rule without the notation, and it is the fastest way to be right out loud.
The relationship
$P(R\mid D) = \frac{P(D\mid R)P(R)}{P(D\mid R)P(R)+P(D\mid C)P(C)} = \frac{0.03 \times 0.6}{0.018+0.004} = \frac{9}{11}$
P(R) the retail share of the book, 60%
P(D|R) the retail default rate, 3%
P(D|C) the corporate default rate, 1%
What it says in wordsThe chance a default came from retail is retail's share of all defaults.
For a risk team this is not an exam trick. The mix of a defaulted pool drives which collections team gets the work, which recovery rate applies and which model needs recalibrating. A 60% share of the book becomes an 82% share of the defaults.
Where candidates lose it
The trap is answering 60%, the share of the book, and ignoring the fact that the loan defaulted. That is base rate thinking run backwards: the prior is treated as the answer.
The other slip is answering 75% by comparing the default rates alone, 3 against 1. That forgets the book is not split evenly. Weight both: 1.8 against 0.4.
What the interviewer asks next
If the book were 50:50, what would the answer be?
Retail LGD is 60% and corporate LGD is 30%. What share of the loss comes from retail?
Two loans default. What is the chance both are retail?
067A forward contract's value to the bank at the one-year horizon is normally distributed with mean zero and standard deviation Rs 10 crore. What is the expected positive exposure to the counterparty at that date?Counterparty exposure and collateralCoreCounterparty riskQuant risk
Try it first
The expected value of the forward is zero. What is the expected exposure?
Show the worked solution
About Rs 4.0 crore, or 0.4 standard deviations. Exposure is the forward's value when it is positive and zero when it is negative. For a normal distribution with mean zero, the expected value of the positive part is sigma divided by the square root of 2 pi, about 0.399 x Rs 10 crore. Half the outcomes contribute nothing; the other half average about Rs 8 crore.
Why is the exposure positive when the expected value is zero?
Lending a friend your bicycle is a one-sided risk: if the friend loses it, you are out a bicycle; if the friend buys you a better one, you gain nothing from the loan itself. Counterparty exposure has the same shape. When the contract is in your favour, a default costs you its value; when it is against you, you still owe the full amount, so negative outcomes count as zero, not as a gain. Cutting off the left half leaves a positive average.
The forward's value in one year is centred on zero with a standard deviation of Rs 10 crore, but only the shaded right half is exposure. Counting the left half as zero gives an expected exposure of about Rs 4.0 crore, 0.4 standard deviations, while the 97.5% potential future exposure is about Rs 19.6 crore.
Where does 0.4 come from, and what is it called?
Integrating v times the normal density from zero to infinity gives sigma times the density at zero, which is 1 over the square root of 2 pi, about 0.399. At a single future date desks usually call this number the expected exposure; averaged over the life of the trade it becomes expected positive exposure, the input to credit valuation adjustment. The tail measure, potential future exposureA high percentile of the exposure distribution at a future date, used to size limits on how much a counterparty could owe. at 97.5%, is 1.96 sigma, about Rs 19.6 crore.
The relationship
$EE = E[\max(V,0)] = \int_0^{\infty} v\,\phi(v)\,dv = \frac{\sigma}{\sqrt{2\pi}} \approx 0.399 \times 10 = 3.99$
V the forward's value to the bank at the horizon
\phi the normal density with mean zero and standard deviation sigma
\sigma Rs 10 crore
What it says in wordsExpected exposure averages only the positive outcomes, which for a zero-mean normal is about 0.4 standard deviations.
The limitation: this assumes no collateral and no netting with other trades. A daily margin agreement would cut the exposure to the move over the few days it takes to call and receive margin, which is far smaller than a year's worth of drift.
Where candidates lose it
The trap is answering zero because the forward is fair and its expected value is zero. Exposure is an option-like quantity: it keeps the upside of the value and discards the downside, so it is always at least zero on average.
The second slip is quoting Rs 8 crore, the average of the positive half alone. That forgets the half of outcomes where exposure is zero. Average across all outcomes: about Rs 4 crore.
What the interviewer asks next
If the forward's expected value were Rs 5 crore instead of zero, would the expected exposure rise by more or less than Rs 5 crore?
How does daily collateral change this number?
Why does exposure on an interest rate swap rise and then fall over its life?
068A Rs 100 crore loan has a 2% probability of default and a 50% loss given default. What is the expected loss, and what is the standard deviation of the loss that capital has to cover?Credit risk arithmeticCoreBank credit risk
Try it first
How does the standard deviation of loss compare with the expected loss?
Show the worked solution
Expected loss is Rs 1 crore; the standard deviation of loss is Rs 7 crore. If the loan defaults, the bank loses 50% of Rs 100 crore, Rs 50 crore; otherwise it loses nothing. The average is 2% x 50 = Rs 1 crore, which the spread should cover. The standard deviation is 50 x the square root of 0.02 x 0.98, which is Rs 7 crore, the unexpected loss capital exists for.
Why is the average loss the wrong thing to hold capital against?
A shopkeeper insuring against a one-in-fifty chance of a Rs 5 lakh fire can budget Rs 10,000 a year for the premium, the average. But in the year the fire happens, the budget is useless; what saves the shop is savings. Expected loss is a cost of doing business and belongs in the price; unexpected lossThe spread of possible losses around the expected loss, usually measured by a standard deviation or a high percentile, which capital is held to absorb. is the surprise, and it is what capital is for.
The loan either loses nothing, 98% of the time, or loses Rs 50 crore, 2% of the time. The expected loss is Rs 1 crore and the standard deviation is Rs 7 crore, so the surprise the bank must be able to absorb is seven times the average it prices in.
How do you get Rs 7 crore quickly?
Treat default as a coin that lands bad with probability p. Its standard deviation is the square root of p times (1 - p): the square root of 0.0196, which is 0.14. Multiply by the loss if it lands bad, Rs 50 crore, and you have Rs 7 crore. The ratio of seven to one is typical for a low-default loan, and it grows as the default probability falls: rarer defaults mean a mean closer to zero and a relatively wider spread.
The relationship
$EL = PD \times LGD \times EAD = 1 \qquad UL = LGD \times EAD \times \sqrt{PD(1-PD)} = 50 \times 0.14 = 7$
PD probability of default, 2%
LGD loss given default, 50%
EAD exposure at default, Rs 100 crore
What it says in wordsThe average loss is the default chance times the loss; the spread is the loss times the square root of p times one minus p.
Say the limitation. For one loan, the standard deviation is a poor summary: the real outcome is 0 or 50, never 7. In a large, diversified portfolio losses spread out and the standard deviation becomes meaningful, and capital models set it at a high percentile of the portfolio loss rather than one standard deviation of a single loan.
Where candidates lose it
The trap is stopping at the expected loss of Rs 1 crore and treating it as the risk. It is the one number the bank should never be surprised by; the risk is the spread.
The second slip is using the variance formula for a continuous variable, or forgetting the square root and quoting Rs 0.98 crore. Say the Bernoulli formula aloud, p times one minus p, then take the root.
What the interviewer asks next
What happens to UL if the default probability falls to 0.5%?
The bank holds 100 such loans, independent of each other. What is the portfolio UL?
Why do defaults in a real portfolio not behave independently?
071Assume Rs 100 lakh crore of government bonds are outstanding with an average modified duration of 7. Roughly how much does their total market value change for a 1 basis point rise in yields?Logic, estimation and brainteasersCoreTreasury and ALMBank market risk
Try it first
Order of magnitude first: what does one basis point cost the holders?
Show the worked solution
About Rs 7,000 crore of value lost for a 1 basis point rise. A bond's price falls by roughly its modified duration times the change in yield. For the whole stock that is Rs 100 lakh crore x 7 x 0.0001, which is 0.07 lakh crore, or Rs 7,000 crore. The same arithmetic makes a 100 basis point rise cost about Rs 7 lakh crore.
Why does a hundredth of a per cent move so much money?
A one paisa rise in the price of petrol means nothing to one driver and a great deal to an oil company selling crores of litres. The move is small; the base is enormous. The DV01The change in value of a bond or portfolio for a one basis point change in yield, sometimes called PV01. of a holding is value times duration times one basis point, so a stock worth Rs 100 lakh crore with a duration of 7 carries Rs 7,000 crore of value per basis point.
Rs 100 lakh crore of bonds times a modified duration of 7 times one basis point is 0.07 lakh crore, which is Rs 7,000 crore of value per basis point. Scaled to a 100 basis point move the change is Rs 7 lakh crore.
How do you keep the units straight under pressure?
Convert once, at the end, and say the conversion aloud. One lakh crore is 1,00,000 crore, so 0.07 lakh crore is 7,000 crore; do the multiplication in the big unit and translate only the answer. A useful check: duration 7 means a 1% move changes value by 7%, so 1 basis point, a hundredth of that, changes it by 0.07%, and 0.07% of 100 lakh crore is 0.07 lakh crore.
The relationship
$\Delta V \approx -D_{\text{mod}} \times V \times \Delta y = -7 \times 100 \times 0.0001 = -0.07 \text{ lakh crore}$
D_mod modified duration, 7
V market value of the bonds, 100 lakh crore, assumed
\Delta y the change in yield, one basis point or 0.0001
What it says in wordsValue change is duration times value times the yield change, with a minus sign because prices fall when yields rise.
State the limits. The stock and duration are assumptions for the puzzle, not current figures, which you would look up. The duration rule ignores convexity, harmless at one basis point but not at a hundred, where the true fall is a little smaller than Rs 7 lakh crore. And the loss lands on whoever holds the bonds, which is why a bank's treasury watches the DV01 of its government bond book daily.
Where candidates lose it
The trap is the units. Candidates multiply correctly to 0.07 and then say Rs 0.07 crore or Rs 70 crore because they lose track of lakh crore. Carry the unit through every step and convert once.
The second slip is forgetting that duration is in per cent per per cent, then dividing by 100 twice. One basis point is 0.0001 as a decimal; multiply by it once.
What the interviewer asks next
If banks hold a third of the stock, what is the banking system's DV01?
Why is the loss for a 100 basis point rise slightly less than 100 times the DV01?
How would a bank hedge part of this exposure?
074Under a normal distribution, 99% VaR should be about 1.41 times 95% VaR. A desk's historical VaR is Rs 5 crore at 95% and Rs 11 crore at 99%. What does the ratio tell you about the desk's P&L?VaR and expected shortfallCoreBank market riskModel validation
Try it first
The desk's ratio is 2.2 against a normal 1.41. What is the most likely reading?
Show the worked solution
The desk's loss tail is much fatter than a normal distribution. For a normal, 2.326 over 1.645 gives a ratio of 1.41, so a 95% VaR of Rs 5 crore implies about Rs 7.07 crore at 99%. The desk shows Rs 11 crore, a ratio of 2.2, fatter even than a Student t with 3 degrees of freedom at 1.93. Rare losses are far larger than volatility alone would suggest.
Why does the ratio say anything about the shape of the tail?
If a city's 1-in-20 rainy day brings 5 cm of rain and its 1-in-100 day brings 11 cm, the storms are of a different kind from the drizzle, not just more of it. Scaling up the whole distribution would move both numbers together. Volatility stretches every quantile by the same factor, so the ratio between two quantiles is a pure measure of shape, and a ratio well above 1.41 means the tail is fatter than normal.
A normal distribution and a fat-tailed one with the same 95% VaR of Rs 5 crore diverge in the tail: the normal's 99% VaR is Rs 7.07 crore and the fat-tailed one's is Rs 11 crore. The desk's ratio of 2.2 exceeds the 1.41 of a normal and the 1.93 of a Student t with 3 degrees of freedom.
What would you do with that finding?
Two things. First, distrust any risk number on this desk that is built from volatility and a normal multiplier, such as a parametric VaR or a stress scaled from the 95% figure, because it will understate the 99% loss by about a third. Second, look at the P&L on the worst days: fat tails on a desk usually come from positions that pay small amounts often and lose large amounts rarely, such as sold options or carry trades.
The relationship
$\frac{\text{VaR}_{99}}{\text{VaR}_{95}}\Big|_{\text{normal}} = \frac{2.326}{1.645} = 1.41 \qquad \frac{11}{5} = 2.2$
2.326, 1.645 the one-sided 99% and 95% points of a standard normal
11, 5 the desk's historical 99% and 95% VaR, Rs crore
What it says in wordsFor a normal the ratio is fixed at 1.41 whatever the volatility, so a larger ratio signals a heavier tail.
The limitation is sample size. A 99% historical VaR from a year of data rests on two or three observations, so one extreme day could produce the 2.2 by itself. Check the ratio across several windows before concluding the desk's business is fat-tailed rather than unlucky.
Where candidates lose it
The trap is reading the ratio as higher volatility. Volatility changes the size of both numbers but not their ratio; only the shape of the distribution changes the ratio.
The second slip is concluding the desk is fine because the 95% number looks normal. A fat tail hides at the 95% level and shows up only further out, which is exactly why regulators moved capital towards expected shortfall.
What the interviewer asks next
What would a ratio below 1.41 suggest?
How would you expect the desk's expected shortfall at 97.5% to compare with a normal's?
Which kinds of trading positions tend to produce fat left tails?

\lambda	the decay factor, 0.94
\sigma_{t-1}	yesterday's volatility estimate, 1.2%
r_t	today's return, minus 3%

g	the compound annual growth rate
\mu	the arithmetic average annual return, 10%
\sigma	the annual volatility, 30%

\sigma_P	the volatility of the unhedged position
\rho	the correlation between position and hedge, 0.8

z	how many standard deviations the threshold sits above the mean
P(Z>2)	the share of a normal distribution beyond two standard deviations on one side

Try it first

Why blend variances rather than volatilities?

What happens next, and what are the model's limits?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the average return not the growth rate?

Where does half the variance come from, and how good is it?

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Why does a correlation of 0.8 leave so much behind?

How do you get from variance left to volatility left?

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Why does a sensible threshold still fire when nothing is wrong?

How precise is the answer of about six?

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Why must call minus put equal the stock minus the discounted strike?

Why would a risk manager care about the rate hidden in option prices?

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Why is the answer not simply 60%?

What is the quickest way to do this in your head?

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Why is the exposure positive when the expected value is zero?

Where does 0.4 come from, and what is it called?

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Why is the average loss the wrong thing to hold capital against?

How do you get Rs 7 crore quickly?

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Why does a hundredth of a per cent move so much money?

How do you keep the units straight under pressure?

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Why does the ratio say anything about the shape of the tail?

What would you do with that finding?

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